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Atomic physics

Importance of Hydrogen Atom

- Hydrogen is the simplest atom
- The quantum numbers used to characterize the

allowed states of hydrogen can also be used to

describe (approximately) the allowed states of

more complex atoms - This enables us to understand the periodic table
- The hydrogen atom is an ideal system for

performing precise comparisons of theory and

experiment - Also for improving our understanding of atomic

structure - Much of what we know about the hydrogen atom can

be extended to other single-electron ions - For example, He and Li2

Early Models of the Atom

- J.J. Thomsons model of the atom
- A volume of positive charge
- Electrons embedded throughout the volume
- A change from Newtons model of the atom as a

tiny, hard, indestructible sphere

watermelon model

Experimental tests

- Expect
- Mostly small angle scattering
- No backward scattering events
- Results
- Mostly small scattering events
- Several backward scatterings!!!

Early Models of the Atom

- Rutherfords model
- Planetary model
- Based on results of thin foil experiments
- Positive charge is concentrated in the center of

the atom, called the nucleus - Electrons orbit the nucleus like planets orbit

the sun

Problem Rutherfords model

The size of the atom in Rutherfords model is

about 1.0 1010 m. (a) Determine the attractive

electrical force between an electron and a proton

separated by this distance. (b) Determine (in

eV) the electrical potential energy of the atom.

The size of the atom in Rutherfords model is

about 1.0 1010 m. (a) Determine the attractive

electrical force between an electron and a proton

separated by this distance. (b) Determine (in eV)

the electrical potential energy of the atom.

Electron and proton interact via the Coulomb force

- Given
- r 1.0 1010 m
- Find
- F
- PE

Potential energy is

Difficulties with the Rutherford Model

- Atoms emit certain discrete characteristic

frequencies of electromagnetic radiation - The Rutherford model is unable to explain this

phenomena - Rutherfords electrons are undergoing a

centripetal acceleration and so should radiate

electromagnetic waves of the same frequency - The radius should steadily decrease as this

radiation is given off - The electron should eventually spiral into the

nucleus - It doesnt

28.2 Emission Spectra

- A gas at low pressure has a voltage applied to it
- A gas emits light characteristic of the gas
- When the emitted light is analyzed with a

spectrometer, a series of discrete bright lines

is observed - Each line has a different wavelength and color
- This series of lines is called an emission

spectrum

Emission Spectrum of Hydrogen

- The wavelengths of hydrogens spectral lines can

be found from - RH is the Rydberg constant
- RH 1.0973732 x 107 m-1
- n is an integer, n 1, 2, 3,
- The spectral lines correspond to
- different values of n
- A.k.a. Balmer series
- Examples of spectral lines
- n 3, 656.3 nm
- n 4, 486.1 nm

Absorption Spectra

- An element can also absorb light at specific

wavelengths - An absorption spectrum can be obtained by passing

a continuous radiation spectrum through a vapor

of the gas - The absorption spectrum consists of a series of

dark lines superimposed on the otherwise

continuous spectrum - The dark lines of the absorption spectrum

coincide with the bright lines of the emission

spectrum

Applications of Absorption Spectrum

- The continuous spectrum emitted by the Sun passes

through the cooler gases of the Suns atmosphere - The various absorption lines can be used to

identify elements in the solar atmosphere - Led to the discovery of helium

Difficulties with the Rutherford Model

- Cannot explain emission/absorption spectra
- Rutherfords electrons are undergoing a

centripetal acceleration and so should radiate

electromagnetic waves of the same frequency, thus

leading to electron falling on a nucleus in

about 10-12 seconds!!!

Bohrs model addresses those problems

28.3 The Bohr Theory of Hydrogen

- In 1913 Bohr provided an explanation of atomic

spectra that includes some features of the

currently accepted theory - His model includes both classical and

non-classical ideas - His model included an attempt to explain why the

atom was stable

Bohrs Assumptions for Hydrogen

- The electron moves in circular orbits around the

proton under the influence of the Coulomb force

of attraction - The Coulomb force produces the centripetal

acceleration - Only certain electron orbits are stable
- These are the orbits in which the atom does not

emit energy in the form of electromagnetic

radiation - Therefore, the energy of the atom remains

constant and classical mechanics can be used to

describe the electrons motion - Radiation is emitted by the atom when the

electron jumps from a more energetic initial

state to a lower state - The jump cannot be treated classically

Bohrs Assumptions

- More on the electrons jump
- The frequency emitted in the jump is related to

the change in the atoms energy - It is generally not the same as the frequency of

the electrons orbital motion - The size of the allowed electron orbits is

determined by a condition imposed on the

electrons orbital angular momentum

Results

- The total energy of the atom
- Newtons law
- This can be used to rewrite kinetic energy as
- Thus, the energy can also be expressed as

Bohr Radius

- The radii of the Bohr orbits are quantized (

) - This shows that the electron can only exist in

certain allowed orbits determined by the integer

n - When n 1, the orbit has the smallest radius,

called the Bohr radius, ao - ao 0.0529 nm

Radii and Energy of Orbits

- A general expression for the radius of any orbit

in a hydrogen atom is - rn n2 ao
- The energy of any orbit is
- En - 13.6 eV/ n2
- The lowest energy state is called the ground

state - This corresponds to n 1
- Energy is 13.6 eV
- The next energy level has an energy of 3.40 eV
- The energies can be compiled in an energy level

diagram - The ionization energy is the energy needed to

completely remove the electron from the atom - The ionization energy for hydrogen is 13.6 eV

Energy Level Diagram

- The value of RH from Bohrs analysis is in

excellent agreement with the experimental value - A more generalized equation can be used to find

the wavelengths of any spectral lines - For the Balmer series, nf 2
- For the Lyman series, nf 1
- Whenever a transition occurs between a state, ni

and another state, nf (where ni gt nf), a photon

is emitted - The photon has a frequency f (Ei Ef)/h and

wavelength

Problem Transitions in the Bohrs model

A photon is emitted as a hydrogen atom undergoes

a transition from the n 6 state to the n 2

state. Calculate the energy and the wavelength of

the emitted photon.

A photon is emitted as a hydrogen atom undergoes

a transition from the n 6 state to the n 2

state. Calculate the energy and the wavelength of

the emitted photon.

- Given
- ni 6
- nf 2
- Find
- l
- Eg

Photon energy is

Bohrs Correspondence Principle

- Bohrs Correspondence Principle states that

quantum mechanics is in agreement with classical

physics when the energy differences between

quantized levels are very small - Similar to having Newtonian Mechanics be a

special case of relativistic mechanics when v ltlt

c

Successes of the Bohr Theory

- Explained several features of the hydrogen

spectrum - Accounts for Balmer and other series
- Predicts a value for RH that agrees with the

experimental value - Gives an expression for the radius of the atom
- Predicts energy levels of hydrogen
- Gives a model of what the atom looks like and how

it behaves - Can be extended to hydrogen-like atoms
- Those with one electron
- Ze2 needs to be substituted for e2 in equations
- Z is the atomic number of the element

Atomic Transitions Energy Levels

- An atom may have many possible energy levels
- At ordinary temperatures, most of the atoms in a

sample are in the ground state - Only photons with energies corresponding to

differences between energy levels can be absorbed

Atomic Transitions Stimulated Absorption

- The blue dots represent electrons
- When a photon with energy E is absorbed, one

electron jumps to a higher energy level - These higher levels are called excited states
- E h E2 E1
- In general, E can be the difference between any

two energy levels

Atomic Transitions Spontaneous Emission

- Once an atom is in an excited state, there is a

constant probability that it will jump back to a

lower state by emitting a photon - This process is called spontaneous emission

Atomic Transitions Stimulated Emission

- An atom is in an excited stated and a photon is

incident on it - The incoming photon increases the probability

that the excited atom will return to the ground

state - There are two emitted photons, the incident one

and the emitted one - The emitted photon is in exactly in phase with

the incident photon

Population Inversion

- When light is incident on a system of atoms, both

stimulated absorption and stimulated emission are

equally probable - Generally, a net absorption occurs since most

atoms are in the ground state - If you can cause more atoms to be in excited

states, a net emission of photons can result - This situation is called a population inversion

Lasers

- To achieve laser action, three conditions must be

met - The system must be in a state of population

inversion - The excited state of the system must be a

metastable state - Its lifetime must be long compared to the normal

lifetime of an excited state - The emitted photons must be confined in the

system long enough to allow them to stimulate

further emission from other excited atoms - This is achieved by using reflecting mirrors

Production of a Laser Beam

Laser Beam He Ne Example

- The energy level diagram for Ne
- The mixture of helium and neon is confined to a

glass tube sealed at the ends by mirrors - A high voltage applied causes electrons to sweep

through the tube, producing excited states - When the electron falls to E2 in Ne, a 632.8 nm

photon is emitted

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